Proof that cross product = Area of parrelelogram

Not sure if I should be in this thread or in the geometry one, but here goes:

The problem that I was given was:
Find the area of a paralellogram spanned by the vectors a = i + j and b = 4i - j
Now prove the following |a X b| =|a||b|sin(theta) for ANY ( 2 dimensional ) vectors a and b where theta is the angle between the vectors.

Now I know that the area of a parrallogram is the magnitude of the cross product, and I can easily show that the area of the parrallelogram is the same as |a||b|sin(theta), but how do you prove that |axb| = area of parrelelogram?

Actually, I would be inclined to look at it the other way- with the cross product of two vectors [b]defined[\b] by and the direction given by the "right hand rule", prove that , but, of course, you can go either way.

In order to show your way, define a "new" product, say *, by , and the "right hand rule". We can easily get the rules , , and as well as , and the general .

Now, for and , multiply "term by term" and use the basic products above to show that this "new" product is precisely the cross product.